The Momentum Conserving Scheme for Two-Layer Shallow Flows

This paper confronts the numerical simulation of steady flows of fluid layers through channels of varying bed and width. The fluid consists of two immiscible fluid layers with constant density, and it is assumed to be of a one-dimensional shallow flow. The governing equation is a coupled system of two-layer shallow water models. In this paper, we apply a direct extension of the momentum conserving scheme previously used for solving the one layer shallow water equations. Computations of various steady-state solutions are used to demonstrate the performance of the proposed numerical scheme. Under the influence of a given flow rate, the numerical steady interface is generated in a channel topography with a hump. The results obtained confirm the analytic steady interface of the two-layer rigid-lid model. Furthermore, the same scheme was used with an additional artificial damping to simulate the maximal exchange flow in channels of varying width. The numerical steady interface agreed well with the analytical steady solutions.

Numerical Simulating Open-Channel Flows with Regular and Irregular Cross-Section Shapes Based on Finite Volume Godunov-Type Scheme

A numerical model based on the Saint-Venant equations (one-dimensional shallow water equations) is proposed to simulate shallow flows in an open channel with regular and irregular cross-section shapes. The Saint-Venant equations are solved by the finite-volume method based on Godunov-type framework with a modified Harten, Lax, and van Leer (HLL) approximate Riemann solver. Cross-sectional area is replaced by water surface level as one of primitive variables. Two numerical integral algorithms, compound trapezoidal and Gauss–Legendre integrations, are used to compute the hydrostatic pressure thrust term for natural streams with arbitrary and irregular cross-sections. The Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and second-order Runge–Kutta methods is adopted to achieve second-order accuracy in space and time, respectively. The performance of the resulting scheme is evaluated by application in rectangular channels, trapezoidal channels, and a natural mountain river. The results are compared with analytical solutions and experimental or measured data. It is demonstrated that the numerical scheme can simulate shallow flows with arbitrary cross-section shapes in practical conditions.

Velocimetry Based on Self-Generated Surface Wave Patterns

This paper introduces an image analysis technique applied to an artificially-created disturbance at the free surface of a moving water body as a means of quantifying the average velocity of the water stream for shallow flows. The disturbance was created by a thin object penetrating the free surface with different submerged distances. A V-shaped wake pattern was created by the object of interest through its variation with the water body velocity, the submergence and shape of the piercing body. The angle of the wake pattern decreased with the increase of the velocity for a depth-based Froude number ranging from 0.15 to 0.96. The proof-of-concept experiments presented in this paper, therefore, are usable to quantify the velocity based on the wake angle only in subcritical flow conditions. The results showed the shape of the wake was only slightly influenced by the shape of the object geometry and its submergence. Observations on various types of surface wakes have been documented before, but it is the conversion of these observations into a relatively inexpensive and robust method to estimate the velocity of the moving body that is deemed innovative.

Adaptation of flux-based solvers to 2D two-layer shallow flows with variable density including numerical treatment of the loss of hyperbolicity and drying/wetting fronts

An important feature of the two-layer shallow flow model is that the resulting system of equations cannot be expressed in conservation-law form. Here, the HLLS and ARoe solvers, derived initially for systems of conservation laws, are reformulated and applied to the two-layer shallow flows in a great variety of problems. Their resulting extension and combination allows us to overcome the loss of the hyperbolic character, ensuring energy or exactly balanced property, guarantees positivity of the solution, and provides a correct drying/wetting advance front without requiring tuning parameters. As a result, in those cases where the rich description of internal and external waves cannot be provided by the ARoe solver, HLLS is applied. Variable density is considered in each layer as a result of a bulk density driven by the mixture of different constituents. A wide variety of test cases is presented confirming the properties of this combination, including exactly balanced scenarios in subcritical and subcritical-transcritical scenarios, dam-break problems over bed variations and wet/dry fronts, non-hyperbolic conditions, transcritical exchange flow with loss of hyperbolicity. Despite the complexity of the test cases presented here, accurate and stable simulations are guaranteed, ensuring positivity of the solution without decreasing the time step.

Experimental study of resonant shallow flows past a lateral cavity: a benchmark test for high-resolution numerical models

<p><span>The study of resonant shallow flows past a lateral cavity is of great relevance due to their interest in civil and environmental engineering [1]. Such flows exhibit the presence of a standing gravity wave, called seiche, which is coupled with the shedding of vortices at the opening of the cavity. A complete understanding of such phenomenon is necessary as it may determine the mass exchange between the main channel and the cavity [2]. </span><span>A better insight into this phenomenon helps to improve the design and implementation of innovative river bank restoration techniques</span><span>. An experimental study of the resonant flow in a laboratory flume with a single lateral cavity is herein presented. Five different flow configurations at a fixed Froude number (Fr=0.8) are considered. The main novelty of the present work is the use of a pioneering non-intrusive experimental technique [3] to measure the water surface at the channel-cavity region. This optical technique offers high resolution 2D data in time and space of the water surface evolution, allowing to determine the relevant features of the seiche oscillation, i.e. spatial distribution of oscillation nodes and anti-nodes, oscillation modes and amplitude of the oscillation. Such data are supplemented with Particle Image Velocimetry measurements to perform a more detailed study of the resonance phenomenon. High-resolution two-dimensional amplitude oscillation maps of the seiche phenomenon are presented for the experimental water depth. Experimental velocity fields inside the cavity are presented and confirm the inherent coupling between the unstable shear layer at the opening of the cavity and the gravity standing wave. The high quality of the experimental data reported in this work makes this data set a suitable benchmark for numerical simulation models in order to evaluate their performance in the resolution of turbulent resonant shallow flows.</span></p><p><span>[1] C. Juez, M. Thalmann, A. J. Schleiss & M. J. Franca, Morphological resilience to flow fluctuations of fine sediment deposits in bank lateral cavities, Advances in Water Resources, 115 (2018) 44-59.</span></p><p><span>[2] I. Kimura & T. Hosoda, Fundamental properties of flows in open channels with dead zone, Journal of Hydraulic Engineering 123 (1997) 98-107.</span></p><p><span>[3] S. Martínez-Aranda, J. Fernández-Pato, D. Caviedes-Voullième, I. García-Palacín & P. García-Navarro, Towards transient experimental water surfaces: a new benchmark dataset for 2D shallow water solvers, Advances in water resources, 121 (2018) 130-149.</span></p>